Wikipedia:Reference desk/Archives/Mathematics/2024 August 14

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August 14

Hypothetical US Senate reform

Suppose that the US Senate is reformed to be lightly weighted by population, with each state having either 1, 2, or 3 senators (and keeping the total number of 100 senators constant). What would be a plausible method for determining which states get 1, 2, or 3? 71.126.57.129 (talk) 03:33, 14 August 2024 (UTC)

Representatives are assigned using a greedy algorithm: First, every state is allocated one representative; then whichever state currently has the highest ratio of population to representatives is given another representative; repeat until out of representatives. One could reasonably use the same algorithm for senators, with a maximum number of senators per state allowed.--Antendren (talk) 06:31, 14 August 2024 (UTC)

(ec) You can think of the states as each being one party in a multi-party system, and of the population of each state as being voters for their own party (voting for the party, not for a specific candidate). Several multi-winner voting systems can be used, specifically party-list proportional representation, tweaked to keep the number of seats for each party in a given 1 to n range – in the question as posed n = 3. One possible way is the D'Hondt method, which will assign the seats one by one to the state that is proportionally the least represented and still has fewer than n seats. As you can see at Party-list proportional representation § Apportionment of party seats, there are many other methods.  --Lambiam 06:36, 14 August 2024 (UTC)

One plausible way is to start moving senators from the smallest states to the largest. So pick the smallest state first, take one of their senators and assign it to the largest. Then repeat with the next smallest and largest. Keep going until you run out of states that are small or large enough for it to make sense. This works as if you require the total to be unchanged, with 50 states and 100 senators, the number of states with 1 and 3 must be equal. --2A04:4A43:90FF:FB2D:F067:D97A:B158:E36B (talk) 12:44, 14 August 2024 (UTC)

One approach is to define what is meant by a fair distribution of Senate seats. A reasonable ansatz is to minimize the sum of seats/population subject to the constraints that there are 100 seats total and each state gets between 1 and 3 seats. This is a variant of the knapsack problem. Tito Omburo (talk) 13:15, 14 August 2024 (UTC)

Another theory-based approach is to minimize total dissatisfaction, assuming that the members of a group will feel less happy if the group is underrepresented. Let, for group $i,$ the quantity $p_{i}$ stand for the fraction the group takes up in the total population, and $s_{i}$ for the fraction of the seats allocated to the group (so $\textstyle {\sum _{i}p_{i}=\sum _{i}s_{i}=1}.$) Ideally, $p_{i}=s_{i}=1$ for all groups, but this is rarely posible, given the constraints. For a model of total dissatisfaction, choose some function $f(p,s)$ that represents the dissatisfaction of a group with population weight $p$ and representation weight $s;$ then $\textstyle {\sum _{i}p_{i}f(p_{i},s_{i})}$ stands for the proportionally weighted combined dissatisfaction. Plausibly, $s\geq p,$ meaning that a group is not underrepresented, should imply that $f(p,s)=0;$ furthermore, for fixed $p,$ the dissatisfaction should weakly monotonically decrease with increasing $s;$ conversely, for fixed $s,$ the dissatisfaction should weakly monotonically increase with increasing $p.$ A possible formula is given by $f(p,s)=(p{\dot {-}}s)^{2},$ using truncated subtraction.  --Lambiam 16:55, 14 August 2024 (UTC)